Calculus

Calculus

1. Derivatives

   • Product Rule
     • (fg)' = f'g+g'f
   • Quotient Rule
     • (f/g)' = (f'g - g'f) / g^2
   • Power Rule for Functions
     • (f^n)' = nf^(n-1)f'
   • Chain Rule
     • dy/dx = (dy/du)(du/dx)
       • for y=y(u) and u=u(x)
     • [f(g(x)]' = f'(g(x))g'(x)
   • Finding 2nd Derivative
     • just take the derivative of the derivative
   • f(x) = e^x, f'(x) = e^x
     • f(x) = ln x, f'(x) =1/x
     • g(x) = loga(x), g'(x) = (1/ln a)(1/x)
       • f'(x) = ln(a)a^x
     • Chain Rule and Logs
       • g(x) = ln(f(x))
         • f'(x)/f(x)
     • ln(x) + 1/x (x')

2. Implicit Differentiation

   • Treat x as x and y as f(x), give derivative of y w/respect to x
   • Differentiate both sides w/respect to x with y as f(x)
     • Chain rule, differentiating terms with y
       • Solve for dy/dx in terms of x and y